A straightforward gravitational path integral calculation implies that closed universes are trivial, described by a one dimensional Hilbert space. Two recent papers by Harlow-Usatyuk-Zhao and Abdalla-Antonini-Iliesiu-Levine have sought to ameliorate this issue by defining special rules to incorporate observers into the path integral. However, the proposed rules are different, leading to differing results for the Hilbert space dimension. Moreover, the former work offers a holographic map realized using a non-isometric code construction to complement their path integral result and clarify its physics. In this work, we propose a non-isometric code that implements the second construction, allowing thorough comparison. Our prescription may be thought of as simply removing the portion of the map that acts on the observer, while preserving the rest, creating an effective holographic boundary at the observer-environment interface. This proposal can be directly applied to general holographic maps for both open and closed universes of any dimension.

United States

Chris Akers

Gracemarie Bueller

Oliver DeWolfe

Kenneth Higginbotham

Johannes Reinking

Rudolph Rodriguez

On observers in holographic maps

While the entanglement entropy of a single subregion in quantum field theory is formally infinite and requires regularization, certain combinations of entropies are perfectly finite in the limit that the regulator is removed, the mutual information being a common example. For generic regulator schemes, such as a holographic calculation with a uniform radial cutoff, these quantities show nontrivial dependence on the regulator at finite values of the cutoff. We investigate a holographic regularization scheme defined in three-dimensional anti-de Sitter space constructed from horocycles, curves in two-dimensional hyperbolic space perpendicular to all geodesics approaching a single point on the boundary, that leads to finite information measures that are totally cutoff independent, even at finite values of the regulator. We describe a broad class of such information measures, and describe how the field theory dual to the horocycle regulator is inherently nonlocal.

Sristy Agrawal

Joshua Levin

Horocycle regulator: Exact cutoff-independence in AdS/CFT

We use hyperbolic tensor networks to construct a holographic map for black hole interiors that adds a notion of locality to the non-isometric codes proposed by Akers, Engelhardt, Harlow, Penington, and Vardhan. We use tools provided by these networks to study the relationship between non-isometries and quantum extremal surfaces behind the horizon. Furthermore, we introduce a limited notion of dynamics for these interior tensor networks based on the qudit models introduced by Akers et al., and study the evolution of quantum extremal surfaces in an evaporating black hole. We also find a tensor network description of a wormhole connecting the black hole interior to the radiation, providing a mechanism for interior states and operators to be encoded in the radiation after the Page time. As a particular case, we construct a tensor network realization of the backwards-forwards maps recently proposed to incorporate non-trivial effective dynamics in dynamical constructions of these non-isometric black hole codes.

Tensor networks for black hole interiors: non-isometries, quantum extremal surfaces, and wormholes

The backwards-forwards map, introduced as a generalization of the non-isometric holographic maps of the black hole interior of Akers, Engelhardt, Harlow, Penington, and Vardhan to include non-trivial dynamics in the effective description, has two possible formulations differing in when the post-selection is performed. While these two forms are equivalent on the set of dynamically generated states — states formed from unitary time evolution acting on well-defined initial configurations of infalling matter — they differ on the generic set of states necessary to describe the apparent world of the infalling observer. We show that while both versions successfully reproduce the Page curve, the version involving post-selection as the final step, dubbed the backwards-forwards-post-selection (BFP) map, has the desirable properties of being non-isometric but isometric on average and providing state-dependent reconstruction of bulk operators, while the other version does not. Thus the BFP map is a suitable non-isometric code describing the black hole interior including interior interactions.

Bulk reconstruction and non-isometry in the backwards-forwards holographic black hole map

We introduce a new holographic map for encoding black hole interiors by including both fundamental and effective dynamics. This holographic map is constructed by evolving a state in the effective, semiclassical gravity description of the interior backwards in time to pull the degrees of freedom outside the black hole, before evolving forwards in time in the fundamental description. We show this “backwards-forwards” map is equivalent to a post-selection map of the type introduced by Akers, Engelhardt, Harlow, Penington, and Vardhan, and in the case of trivial effective interactions reduces to their model, while providing a suitable generalization when those interactions are nontrivial. We show the map is equivariant with respect to time evolution, and independent of any interactions outside the black hole. This construction includes interactions with an infaller in a way that preserves the unitarity of black hole evolution exactly and does not allow for superpolynomial computational complexity.

Non-isometric codes for the black hole interior from fundamental and effective dynamics

We discuss how electromagnetically dualizing a 1-form to a 2-form in <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow></math> exchanges regular and alternate boundary conditions, and thus gauges the originally global <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> symmetry in the dual field theory. The generalized symmetry current dual to the 2-form in the bulk is identified as the dual field strength of the gauged <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math>, and the associated double-trace operator with a logarithmically running coupling is just the gauged <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> Maxwell action. Applying this dualization to an AdS Maxwell-Chern-Simons theory dual to a global <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> model with a ’t Hooft anomaly results in a theory with a modified field strength that holographically realizes a 2-group symmetry. We explicitly carry out the holographic renormalization to verify this, and discuss the generalization to other rank fields in other dimensions.